The present invention is related to a process for calculating the surface displacement due to arbitrary vertical loads on the layered elastic half-space using a fast algorithm.
Evaluation of layered structures comes from the fact that structures with multiple layers are everywhere. Examples include superlattices in solid-state physics (Bimberg et al., 1999), composites in material science (Agarwal and Broutman, 1980), earth or foundation structures (or locally ground foundation) (Murthy, 2003), and layered pavements in highway transportation (Khazanovich, 1994). This in turn has motivated the corresponding analytical and numerical research aimed at both forward and inverse problems associated with layered structures (Wang et al., 2003; Pan and Han, 2005). However, even for the forward problems, namely the elastic response due to surface loading of the layered half-space, no fast method is available which can model complex surface loads.    The following references are background information and are incorporated herein by reference in their entirety:    Agarwal, B. D. and Broutman, L. J. (1980): Analysis and Performance of Fiber Composites. John Wiley & Sons, New York.    Balaban, N. Q., Schwarz, U.S., Riveline, D., Goichberg, P., Tzur, G., Sabanay, I., Mahalu, D., Safran, S., Bershadsky, A., Addadi, L., and Geiger, B. (2001): Force and focal adhesion assembly: a close relationship studied using elastic micropatterned substrates. Nature Cell Biology, 3, 466-472.    Becker, J. M. and Bevis, M. (2004): Love's problem. Geophys. J. Int. 156, 171-178.    Bevis, M., Kendrick, E., Cser, A., and Smalley, R. (2004): Geodetic measurement of the local elastic response to the changing mass of water in Lago Laja, Chile. Phys. Earth Planet. Inter. 141, 71-78.    Bimberg, D., Grundmann, M., and Ledentsov, N. N. (1999): Quantum Dot Heterostructures. John Wiley & Sons, New York.    Chave, A. D. (1983): Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48, 1671-1686.    de Boor, C. R. (1979): A Practical Guide to Splines, Springer-Verlag, New York.    Fukahata, Y. and Matsu'ura, M. (2005): General expressions for internal deformation fields due to a dislocation source in a multilayered elastic half-space. Geophys. J. Int. 161, 507-521.    Gilbert, F. and Backus, G. (1966): Propagator matrices in elastic wave and vibration problems. Geophysics 31, 326-332.    Graig, R. F. (1997): Soil Mechanics. Taylor & Francis Group, London.    Khazanovich, L. (1994): Structural Analysis of Multi-Layered Concrete Pavement Systems. Ph.D. Thesis, University of Illinois, Urbana, Ill.    Love, A. E. H. (1944): A Treatise on the Mathematical Theory of Elasticity. Fourth Edition, Dover Publications, New York.    Lucas, S. K. (1995): Evaluation infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64, 269-282.    Lucas, S. K. and Stone, H. A. (1995): Evaluating infinite integrals involving Bessel functions of arbitrary order. J. Comput. Appl. Math. 64, 217-231.    Murthy, V. N. S. (2003): Geotechnical Engineering: Principles and Practices of Soil Mechanics and Foundation Engineering. Marcel Dekker, Inc., New York.    Pan, E. (1989a): Static response of a transversely isotropic and layered half-space to general dislocation sources. Phys. Earth Planet. Inter. 58, 103-117.    Pan, E. (1989b): Static response of a transversely isotropic and layered half-space to general surface loads. Phys. Earth Planet. Inter. 54, 353-363.    Pan, E. (1997): Static Green's functions in multilayered half-spaces. Applied Mathematical Modelling 21, 509-521.    Pan, E. and Han, F. (2004): Green's functions for transversely isotropic piezoelectric multilayered half-spaces. J. Eng. Math. 49: 271-288.    Pan, E. and Han, F. (2005): Green's functions for transversely isotropic piezoelectric functionally graded multilayered half spaces. International Journal of Solids and Structures, 42, 3207-3233.    Rice, J. R. (1983): Numerical Methods, Software, and Analysis: IMSL Reference Edition. McGraw-Hill Book Company, New York.    Wang, C. D., Tzeng, C. S., Pan, E. and Liao, J. J. (2003): Displacements and stresses due to a vertical point load in an inhomogeneous transversely isotropic half-space. Int. J. Rock Mech. Min. Sci. 40, 667-685.    Watson, G. N. (1996): A Treatise on the Theory of Bessel Functions. Cambridge University Press. Second Edition, Reprinted 1996.    Yue, Z. Q. and Yin, J. H. (1998): Backward transfer-matrix method for elastic analysis of layered solids with imperfect bonding. J. Elasticity 50, 109-128.